Let $W$ be a standard one dimensional Brownian motion, and $\mathcal F_t$ its natural filtration. Consider the SDE

$$dX_t = \mu_X (t, \omega) \, dt + \sigma_X (t, \omega) \, dW_t$$

$$dY_t = \mu_Y (t, \omega) \, dt + \sigma_Y (t, \omega) \, dW_t$$

$$X_0 = x_0, Y_0 = y_0 \text{ a.s.}$$

where $\mu_X, \mu_Y, \sigma_X, \sigma_Y \geq 0$ are progressively measurable with respect to $\mathcal F_t$, and $x_0, y_0$ are constants.

Assume the existence of a solution to the above two equations up to a determinstic time $T$.

**Question:** Suppose $\sigma_X \neq \sigma_Y$ on a subset of $\Omega \times [0, T]$ of positive measure. Then is it true that

$$\mathbb P(Y_T > X_T) > 0?$$